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Impossibility and Manipulability. Section 9.3 and Chapter 10. Review of Conditions and Criteria. There are many conditions and criteria that are used to determine if an election is “fair”

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Impossibility and Manipulability

Section 9.3 and Chapter 10


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Review of Conditions and Criteria

  • There are many conditions and criteria that are used to determine if an election is “fair”

  • These criteria often refer to voters changing their ballots in some way, and the result of the election changing (or not)


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Condorcet Winner Criterion

  • A voting system is said to satisfy CWC provided that, for every possible sequence of preference list ballots, either (1) there is no Condorcet winner or (2) the voting system produces exactly the same winner for this election as does Condorcet’s method.

  • Plurality does not satisfy this criterion, but Condorcet’s method does


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Independence of Irrelevant Alternatives

  • A voting system is said to satisfy IIA if it is impossible for a candidate A to move from nonwinner status to winner status unless at least one voter reverses the order in which he or she had A and the winning candidate ranked.

  • The Borda count does not satisfy IIA, but Condorcet’s method does


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The Pareto Condition

  • If everyone prefers one candidate (say, B) to another candidate (say, D), then this latter candidate (D) should not be among the winners of the election.

  • Plurality satisfies the Pareto condition, but sequential pairwise voting does not


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Monotone

  • If a candidate is a winner, and a new election is held in which the only ballot change made is for some voter to move the former winning candidate higher on his or her ballot, then the original winner should remain a winner.

  • Plurality is monotone, but the Hare system is not


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The Search for a Perfect Voting System

  • All of the methods we have discussed are flawed in some way

  • Why didn’t I just tell you about the “best” voting system in the first place?

  • Recall May’s Theorem says that majority rule is the “best” method for deciding the winner of an election with two candidates


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Arrow’s Impossibility Theorem

  • Named after Kenneth Arrow, an American economist

  • Essentially, the theorem states that there is no perfect voting method

  • It doesn’t just say that we haven’t thought of a perfect system yet; it says that we can never create one


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Arrow’s Impossibility Theorem

  • With three or more candidates and any number of votes, there does not exist (and will never exist) a voting system that:

    • always produces a winner

    • satisfies the Pareto condition

    • satisfies the independence of irrelevant alternatives condition

    • is not a dictatorship


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Proving the Theorem

  • Arrow’s Theorem is hard to prove (he earned a Nobel prize in 1972 for his work in this area)

  • We will prove a weaker version of his theorem

  • To prove that it is impossible to create a voting system, we will assume that we have created such a system

  • This assumption will lead to an impossibility


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Weak Version of Arrow’s Theorem

  • With three or more candidates, there does not exist (and will never exist) a voting system that:

    • satisfies the Condorcet winner criterion

    • satisfies the independence of irrelevant alternatives condition

    • always produces at least one winner


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A Hypothetical Voting System

  • Let’s assume that we have a hypothetical voting system that

    • satisfies the Condorcet winner criterion

    • satisfies the independence of irrelevant alternatives condition

    • always produces at least one winner

  • In other words, we’re assuming that we have exactly the kind of voting system that Arrow’s Theorem says should not exist


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A Problematic Profile

  • Consider this voter profile

    • Since our system “always produces at least one winner,” we might wonder who the winner should be

    • We will show that A is not the winner


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A New Profile

  • What about this profile?

  • Since C is the Condorcet winner, and our hypothetical system satisfies the Condorcet winner criterion, C must be the winner using our hypothetical system also


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Modifying the Profile

  • Now we’ll modify the second profile to turn it into the first one:

  • Notice that the only change was for the second voter to change from C > B > A to B > C > A

  • B is irrelevant to the question of A versus C, so since C was the previous winner and A was a previous non-winner, IIA means that A must continue to be a non-winner


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The Problematic Profile

  • So we have just shown that for this profile, A is not a winner

  • A similar argument shows that B and C are also non-winners

  • But we assumed that our hypothetical system always finds a winner

  • Therefore our hypothetical system can’t exist


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Summary of Chapter 9

  • One “best” way to determine the winner of an election with two candidates: majority rule

  • Many ways to determine the winner of an election with more than two candidates

  • All of these methods are “unfair” in some way

  • We use criteria to be very specific about the ways in which the methods are “unfair”

  • It is impossible to find a completely fair system


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Chapter 10: Manipulability

  • Sometimes, in order to achieve the election result you prefer, you submit a ballot that misrepresents your actual preferences

  • This type of strategic voting is called manipulation, and the misrepresented ballot is referred to as an insincere or disingenuous ballot


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An Example

  • Consider this voter profile, with just two voters

  • If we use the Borda count to determine the winner, B wins with 5 points

  • Assuming that Voter #1 knew the ballot that Voter #2 was going to submit, could Voter #1 have submitted her ballot so that A wins?


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A Manipulated Outcome

  • What if Voter #1 changes her ballot like this:

  • This ballot is insincere: #1 likes B better than C or D, but she has ranked B last to try to change the result

  • Using this new ballot, A is now the Borda count winner

  • Voter #1 prefers this outcome according to her original ballot


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Manipulability

  • A voting system is said to be manipulable if there are two sets of ballots and a voter (we’ll call him “Bob”) such that

    • neither election ends in a tie

    • the only difference between the two sets of ballots is Bob’s ballot

    • Bob prefers (according to his actual preferences as expressed in the first election) the outcome of the second election to that of the first


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Majority Rule

  • In an election with two candidates (A and B), is majority rule manipulable?

  • In order to manipulate the election, A would have to be the winner, but your true preference would have to be B > A

  • The only change you can make to your ballot is to change it to A > B

  • Since May’s Theorem guarantees that majority rule is monotone, changing your vote from a vote for the loser to a vote for the winner cannot change the outcome


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Condorcet’s Method

  • Condorcet’s Method is also non-manipulable

  • If you prefer B, but the winner is A, then A beats B head-to-head even with your vote preferring B over A

  • No matter how you change your ballot, A will still beat B head-to-head


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The Perfect System?

  • Condorcet’s Method has some very nice properties

    • elections never result in ties (assuming the number of voters is odd)

    • satisfies the Pareto condition

    • non-manipulable

    • not a dictatorship

  • However, Condorcet’s Method also sometimes doesn’t produce a winner

  • Is there a “perfect” system that satisfies all of these properties and always gives a winner?


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The Gibbard-Satterthwaite Theorem

  • With three or more candidates and any number of voters, there does not exist (and never will exist) a voting system that always produces a winner, never has ties, satisfies the Pareto condition, is non-manipulable, and is not a dictatorship.


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